Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization
The population balance equations (PBEs) serve as the primary governing equations for simulating the crystallization process. Two-dimensional (2D) PBEs pertain to crystals that exhibit anisotropic growth, which is characterized by changes in two internal coordinates. Because PBEs are the hyperbolic e...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-02-01
|
| Series: | Crystals |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-4352/14/3/234 |
| _version_ | 1797241495806279680 |
|---|---|
| author | Cengceng Dong Chunlei Ruan |
| author_facet | Cengceng Dong Chunlei Ruan |
| author_sort | Cengceng Dong |
| collection | DOAJ |
| description | The population balance equations (PBEs) serve as the primary governing equations for simulating the crystallization process. Two-dimensional (2D) PBEs pertain to crystals that exhibit anisotropic growth, which is characterized by changes in two internal coordinates. Because PBEs are the hyperbolic equations, it becomes imperative to establish a high-resolution scheme to reduce numerical diffusion and numerical dispersion, thereby ensuring accurate crystal size distribution. This paper uses Euler’s first-order explicit (EE) method–Peridynamic Differential Operator (PDDO) to solve 2D PBE, namely, the EE method for discretizing the time derivative and the PDDO for discretizing the internal crystal-size derivative. Five examples, including size-independent growth with smooth and non-smooth distributions, size-dependent growth, nucleation, and size-independent/dependent growth for batch crystallization are considered. The results show that the EE–PDDO method is more accurate than the HR method and that it is as good as the fifth-order Weighted Essential Non-Oscillatory (WENO) method in solving 2D PBE. This study extends the EE–PDDO method to the simulation of 2D PBE, and the advantages of the EE-PDDO method in dealing with discontinuous and sharp front problems are demonstrated. |
| first_indexed | 2024-04-24T18:24:14Z |
| format | Article |
| id | doaj.art-205a203c486347c3a30755233b6b3c60 |
| institution | Directory Open Access Journal |
| issn | 2073-4352 |
| language | English |
| last_indexed | 2024-04-24T18:24:14Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Crystals |
| spelling | doaj.art-205a203c486347c3a30755233b6b3c602024-03-27T13:32:26ZengMDPI AGCrystals2073-43522024-02-0114323410.3390/cryst14030234Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in CrystallizationCengceng Dong0Chunlei Ruan1School of Mathematics & Statistics, Henan University of Science & Technology, Luoyang 471023, ChinaSchool of Mathematics & Statistics, Henan University of Science & Technology, Luoyang 471023, ChinaThe population balance equations (PBEs) serve as the primary governing equations for simulating the crystallization process. Two-dimensional (2D) PBEs pertain to crystals that exhibit anisotropic growth, which is characterized by changes in two internal coordinates. Because PBEs are the hyperbolic equations, it becomes imperative to establish a high-resolution scheme to reduce numerical diffusion and numerical dispersion, thereby ensuring accurate crystal size distribution. This paper uses Euler’s first-order explicit (EE) method–Peridynamic Differential Operator (PDDO) to solve 2D PBE, namely, the EE method for discretizing the time derivative and the PDDO for discretizing the internal crystal-size derivative. Five examples, including size-independent growth with smooth and non-smooth distributions, size-dependent growth, nucleation, and size-independent/dependent growth for batch crystallization are considered. The results show that the EE–PDDO method is more accurate than the HR method and that it is as good as the fifth-order Weighted Essential Non-Oscillatory (WENO) method in solving 2D PBE. This study extends the EE–PDDO method to the simulation of 2D PBE, and the advantages of the EE-PDDO method in dealing with discontinuous and sharp front problems are demonstrated.https://www.mdpi.com/2073-4352/14/3/234Peridynamic Differential Operatorpopulation balance equationcrystallizationanisotropic |
| spellingShingle | Cengceng Dong Chunlei Ruan Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization Crystals Peridynamic Differential Operator population balance equation crystallization anisotropic |
| title | Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization |
| title_full | Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization |
| title_fullStr | Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization |
| title_full_unstemmed | Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization |
| title_short | Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization |
| title_sort | euler s first order explicit method peridynamic differential operator for solving two dimensional population balance equations in crystallization |
| topic | Peridynamic Differential Operator population balance equation crystallization anisotropic |
| url | https://www.mdpi.com/2073-4352/14/3/234 |
| work_keys_str_mv | AT cengcengdong eulersfirstorderexplicitmethodperidynamicdifferentialoperatorforsolvingtwodimensionalpopulationbalanceequationsincrystallization AT chunleiruan eulersfirstorderexplicitmethodperidynamicdifferentialoperatorforsolvingtwodimensionalpopulationbalanceequationsincrystallization |